Question

The position of a moving particle is given as a function of time \(t\) to be $$\mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin (\omega t)+\hat{\mathbf{z}} v_{\mathrm{o}} t$$ where \(b, c, v_{\mathrm{o}}\) and \(\omega\) are constants. Describe the particle's orbit.

Short answer

The particle moves in a helical path with elliptical cross-section defined by \( b \) and \( c \), advancing linearly with speed \( v_{\mathrm{o}} \) along the z-axis.

Step-by-step solution

Interpretation of the Position Function

The given position function \( \mathbf{r}(t) = \hat{\mathbf{x}} b \cos(\omega t) + \hat{\mathbf{y}} c \sin(\omega t) + \hat{\mathbf{z}} v_{\mathrm{o}} t \) describes a particle's position in 3D space over time. The components \( \hat{\mathbf{x}} b \cos(\omega t) \) and \( \hat{\mathbf{y}} c \sin(\omega t) \) describe circular motion in the plane perpendicular to the \( \hat{\mathbf{z}} \)-axis, while \( \hat{\mathbf{z}} v_{\mathrm{o}} t \) describes linear motion along the \( \hat{\mathbf{z}} \)-axis.

Analyzing the Motion in the XY-plane

The \( x \)- and \( y \)-components of the position vector involve \( \cos(\omega t) \) and \( \sin(\omega t) \), respectively, with coefficients \( b \) and \( c \), indicating that the motion in the \( xy \)-plane is elliptical. The equation for an ellipse centered at the origin and in the \( xy \)-plane is \( \frac{x^2}{b^2} + \frac{y^2}{c^2} = 1 \). This confirms the motion is elliptical in the plane with semi-major axis \( b \) and semi-minor axis \( c \).

Examining the Motion Along the Z-axis

The \( z \)-component is linear: \( \hat{\mathbf{z}} v_{\mathrm{o}} t \), meaning the particle moves at a constant speed \( v_{\mathrm{o}} \) along the \( z \)-axis. The motion is not bounded, and as time increases, the z-coordinate increases linearly.

Combining the Components for Spiral Motion

Incorporating both the elliptical motion in the \( xy \)-plane and the linear motion along the \( z \)-axis, the particle follows a helical or spiral path. The radius of the helical path is characterized by the ellipse defined by \( b \) and \( c \), while the pitch of the spiral depends on \( v_{\mathrm{o}} \).

Key concepts

Elliptical Motion

Elliptical motion is a type of motion where the path taken is in the shape of an ellipse. In a 2D plane, particularly in the XY-plane, this can be represented mathematically by the equation \( \frac{x^2}{b^2} + \frac{y^2}{c^2} = 1 \). Here, \( b \) and \( c \) are constants that determine the lengths of the semi-major and semi-minor axes of the ellipse, respectively.
Elliptical motion is often seen in the natural paths of celestial bodies, such as planets orbiting around a star. This motion is characterized by a periodic nature, as the parameters \( \cos(\omega t) \) and \( \sin(\omega t) \) cycle between -1 and 1. This cycling allows for a smooth, continuous motion along the elliptical path.
Another key feature of elliptical motion is its symmetrical properties, making it a much-studied concept in physics and engineering. Understanding elliptical motion allows for the analysis of many real-world phenomena.

Circular Motion

Circular motion is a specialized case of elliptical motion where the semi-major and semi-minor axes are equal, making the path a perfect circle. In the context of the exercise, if \( b = c \), the elliptical path in the XY-plane becomes circular.
Circular motion is uniform if the speed is constant, meaning the particle moves with a steady speed along the circular path. This type of motion is seen in many applications, such as in the rotation of wheels or the orbits of celestial bodies where gravity provides a centripetal force.
In circular motion, the direction of the velocity changes continuously, even if the speed remains constant. This necessitates a constant centripetal force directed towards the center of the circle to maintain the path of the motion.

Linear Motion

Linear motion refers to movement along a straight line, which is characterized by a constant velocity, meaning both speed and direction remain unchanged. In the given problem, the component \( \hat{\mathbf{z}} v_{\mathrm{o}} t \) describes linear motion along the Z-axis.
This straightforward path is the simplest form of motion and serves as the building block for understanding more complex trajectories.
Linear motion can become more complex when other forces or influences act upon the particle, yet in this scenario, it remains uniform, as velocity \( v_{\mathrm{o}} \) is constant. Linear motion is pivotal in various fields like kinematics and is often the starting point in studying motion.

Spiral Path

A spiral path, or helical path, involves motion where an object travels in circles while also moving along an axis, creating a three-dimensional spiral. This motion can be visualized as combining elliptical motion in one plane with linear motion along an axis perpendicular to that plane.
In the exercise, while the particle exhibits elliptical motion in the XY-plane, it also moves linearly along the Z-axis due to \( \hat{\mathbf{z}} v_{\mathrm{o}} t \). Thus, the particle's trajectory spirals upwards or downwards, depending on the sign of \( v_{\mathrm{o}} \).
The spiral nature of the path is significant in various applications, like DNA molecules' structure or certain types of staircases, which make them compact and space-efficient. Understanding spiral motion helps in visualizing complex dynamics in 3D space.

3D Space Trajectory

3D space trajectory is a term used to describe the path of an object moving through three dimensions. Unlike motion confined to a 2D plane, a 3D trajectory provides a comprehensive understanding of an object’s position as it moves with respect to time.
In the given problem, the trajectory is composed of both elliptical motion in the XY-plane and linear motion along the Z-axis, resulting in a 3D spiral pathway. Such trajectories are prevalent in scenarios where particles or objects are not restricted to a single plane, enabling more dynamic movements.
Analyzing 3D trajectories is crucial in fields like aerospace engineering and robotics, as it allows for designing paths that account for all spatial directions. This comprehensive motion analysis enhances navigation and control in a multitude of applications.